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Analyzing midpoint subdivision. (English) Zbl 1236.65017
The midpoint subdivision schemes form a class of subdivision schemes for arbitrary two-manifold meshes. It is observed that the midpoint subdivision surfaces are spline surfaces except for finitely many extraordinary points, which make the analysis of smoothness more difficult. D. Zorin and P. Schröder [Comput. Aided Geom. Des. 18, No. 5, 429–454 (2001; Zbl 0969.68155)] proved C 1 smoothness of midpoint subdivision surfaces of degree 2 to 9. The authors develop here a geometric framework, which enables them to prove C 1 continuity of midpoint subdivision surfaces of any degree greater than 1.
MSC:
65D18Computer graphics, image analysis, and computational geometry
References:
[1]Catmull, E.; Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes, Computer-aided design 10, No. 6, 350-355 (1978)
[2]Chen, Q., 2005. Gestalt von Unterteilungsflächen in Ausnahmepunkten. Diplomarbeit, Universität Karlsruhe (TH).
[3]Doo, D. W. H.; Sabin, M. A.: Behaviour of recursive division surfaces near extraordinary points, Computer-aided design 10, No. 6, 356-360 (1978)
[4]Horn, R. A.; Johnson, C. R.: Matrix analysis, (1985)
[5]Micchelli, C. A.; Prautzsch, H.: Uniform refinement of curves, Linear algebra and its applications 114/115, 841-870 (1989) · Zbl 0668.65011 · doi:10.1016/0024-3795(89)90495-3
[6]Peters, J.; Reif, U.: Analysis of algorithms generalizing B-spline subdivision, SIAM J. Numer. anal. 35, No. 2, 728-748 (1998) · Zbl 0913.65011 · doi:10.1137/S0036142996304346
[7]Peters, J.; Reif, U.: Subdivision surfaces, (2008)
[8]Prautzsch, H.: Smoothness of subdivision surfaces at extraordinary points, Advances in computational mathematics 9, 377-389 (1998) · Zbl 0918.65094 · doi:10.1023/A:1018945708536
[9]Qu, R., 1990. Recursive subdivision algorithms for curve and surface design. PhD thesis, Brunel University.
[10]Reif, U.: A unified approach to subdivision algorithms near extraordinary vertices, Computer aided geometric design 12, No. 2, 153-174 (1995) · Zbl 0872.65007 · doi:10.1016/0167-8396(94)00007-F
[11]Zorin, D. N.; Schröder, P.: A unified framework for primal/dual quadrilateral subdivision schemes, Computer aided geometric design 18, No. 5, 429-454 (2001) · Zbl 0969.68155 · doi:10.1016/S0167-8396(01)00040-1